Improved Bounds on the Space Complexity of Circuit Evaluation
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912441162530816 |
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| author | Shalunov, Yakov |
| author_facet | Shalunov, Yakov |
| contents | Williams (STOC 2025) recently proved that time-$t$ multitape Turing machines can be simulated using $O(\sqrt{t \log t})$ space using the Cook-Mertz (STOC 2024) tree evaluation procedure. As Williams notes, applying this result to fast algorithms for the circuit value problem implies an $O(\sqrt{s} \cdot \mathrm{polylog}\; s)$ space algorithm for evaluating size $s$ circuits.
In this work, we provide a direct reduction from circuit value to tree evaluation without passing through Turing machines, simultaneously improving the bound to $O(\sqrt{s \log s})$ space and providing a proof with fewer abstraction layers.
This result can be thought of as a "sibling" result to Williams' for circuit complexity instead of time; in particular, using the fact that time-$t$ Turing machines have size $O(t \log t)$ circuits, we can recover a slightly weakened version of Williams' result, simulating time-$t$ machines in space $O(\sqrt{t} \log t)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20950 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved Bounds on the Space Complexity of Circuit Evaluation Shalunov, Yakov Computational Complexity Williams (STOC 2025) recently proved that time-$t$ multitape Turing machines can be simulated using $O(\sqrt{t \log t})$ space using the Cook-Mertz (STOC 2024) tree evaluation procedure. As Williams notes, applying this result to fast algorithms for the circuit value problem implies an $O(\sqrt{s} \cdot \mathrm{polylog}\; s)$ space algorithm for evaluating size $s$ circuits. In this work, we provide a direct reduction from circuit value to tree evaluation without passing through Turing machines, simultaneously improving the bound to $O(\sqrt{s \log s})$ space and providing a proof with fewer abstraction layers. This result can be thought of as a "sibling" result to Williams' for circuit complexity instead of time; in particular, using the fact that time-$t$ Turing machines have size $O(t \log t)$ circuits, we can recover a slightly weakened version of Williams' result, simulating time-$t$ machines in space $O(\sqrt{t} \log t)$. |
| title | Improved Bounds on the Space Complexity of Circuit Evaluation |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2504.20950 |